6 Computational Mechanics of Composite Materials if only the Lesbegue integral with respect to the probabilistic measure exists and converges. Lemma Eel=e (1.21) Lemma There holds for any random numbers X and the real numbers c,E (1.22) Lemma There holds for any independent random variables X ex.-fex.] (1.23) Definition Let us consider the following random variable X:defined on the probabilistic space (F,P).The variance of the variable X is defined as Var(X)=[(X(@)-E[X]YdP(@) (1.24) Q and the standard deviation is called the quantity G(X)=Var(X) (1.25) Lemma 女Var(c)=0 (1.26 Lemma Y Var(cX)=c-Var(X) (1.27) E Lemma There holds for any two independent random variables X and Y Var(X +Y)=Var(X)+Var(Y) (1.28) Var(X.Y)=E[X].Var(Y)+Var(X).Var(Y)+Var(X).E[Y] (1.29)6 Computational Mechanics of Composite Materials if only the Lesbegue integral with respect to the probabilistic measure exists and converges. Lemma E c c c ∀ = ∈ℜ [ ] (1.21) Lemma There holds for any random numbers Xi and the real numbers ci ∈ℜ ∑ ∑ [ ] = = =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ n i i i n i E ci Xi c E X 1 1 (1.22) Lemma There holds for any independent random variables Xi ∏ ∏ [ ] = ⎥ = ⎦ ⎤ ⎢ ⎣ ⎡ = n i i n i E X i E X 1 1 (1.23) Definition Let us consider the following random variable X :Ω → ℜ defined on the probabilistic space ( ) Ω, F, P . The variance of the variable X is defined as ( ) [ ] ∫ Ω ( ) = ( ) − ( ) 2 Var X X ω E X dP ω (1.24) and the standard deviation is called the quantity σ (X) = Var(X ) (1.25) Lemma ∀ ( ) = 0 ∈ℜ Var c c (1.26) Lemma ( ) ( ) 2 Var cX c Var X c ∀ = ∈ℜ (1.27) Lemma There holds for any two independent random variables X and Y Var( ) X ± Y = Var(X) +Var(Y) (1.28) ( ) [ ] ( ) ( ) ( ) ( ) [ ] 2 2 Var X ⋅Y = E X ⋅Var Y +Var X ⋅Var Y +Var X ⋅ E Y (1.29)